\documentstyle{article} \begin{document} \section*{December 11, 1996 Addendum to: Nakauchi, N., {\em A remark on $\infty$-harmonic functions on Riemannian manifolds}, Electr. J. Diff. Eqns. Vol. 1995(1995), No. 07, pp. 1--10.} In this article, Theorem 1 follows from general properties of the Riemannian curvature tensor, and Corollary 1 is incorrect. The Bochner formula does not seem to work in this situation. \medskip Lemma 1 can be used in proving the following Liouville theorem for ${\rm C}^{3}$-solutions. \medskip \noindent{\bf Theorem A.} {\it Let $M$ be a complete noncompact Riemannian manifold of nonnegative (sectional) curvature. Let $u$ be a bounded $\infty$-harmonic function of ${\rm C}^{3}$-class on $M$. Then $u$ is a constant function. } \medskip The curvature assumption in Theorem A is necessary only for applying the Hessian comparison theorem in the proof (Here we use the operator ${\rm Q}^{ij} = g^{ip}g^{jq}\nabla_{p}\nabla_{q}$). Theorem A also follows from arguments in [Cheng, S.Y., {\em Liouville theorem for harmonic maps}, Proc. Symp. Pure Math. 36(1980), 147-151]. See also the article [Hong, N.C., {\it Liouville theorems for exponentially harmonic functions on Riemannian manifolds}, Manuscripta Math. 77(1992), 41-46]. \bigskip Sincerely yours, Nobumitsu Nakauchi \end{document}